↑

Properties

Label 22.6.33059881728...0000.1
Degree $22$
Signature $[6, 8]$
Discriminant $2^{44}\cdot 3^{17}\cdot 5^{36}$
Root discriminant $130.17$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T27

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1866411072, -1271386368, 370003392, 217405440, -596640480, 458745408, -101665472, -105917632, 58622580, -11080640, 850628, 2140848, 74008, -287440, 15180, -2536, -8096, 24, 980, 40, -39, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 - 39*x^20 + 40*x^19 + 980*x^18 + 24*x^17 - 8096*x^16 - 2536*x^15 + 15180*x^14 - 287440*x^13 + 74008*x^12 + 2140848*x^11 + 850628*x^10 - 11080640*x^9 + 58622580*x^8 - 105917632*x^7 - 101665472*x^6 + 458745408*x^5 - 596640480*x^4 + 217405440*x^3 + 370003392*x^2 - 1271386368*x + 1866411072)
 
gp: K = bnfinit(x^22 - 4*x^21 - 39*x^20 + 40*x^19 + 980*x^18 + 24*x^17 - 8096*x^16 - 2536*x^15 + 15180*x^14 - 287440*x^13 + 74008*x^12 + 2140848*x^11 + 850628*x^10 - 11080640*x^9 + 58622580*x^8 - 105917632*x^7 - 101665472*x^6 + 458745408*x^5 - 596640480*x^4 + 217405440*x^3 + 370003392*x^2 - 1271386368*x + 1866411072, 1)
 

Normalized defining polynomial

\( x^{22} - 4 x^{21} - 39 x^{20} + 40 x^{19} + 980 x^{18} + 24 x^{17} - 8096 x^{16} - 2536 x^{15} + 15180 x^{14} - 287440 x^{13} + 74008 x^{12} + 2140848 x^{11} + 850628 x^{10} - 11080640 x^{9} + 58622580 x^{8} - 105917632 x^{7} - 101665472 x^{6} + 458745408 x^{5} - 596640480 x^{4} + 217405440 x^{3} + 370003392 x^{2} - 1271386368 x + 1866411072 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33059881728000000000000000000000000000000000000=2^{44}\cdot 3^{17}\cdot 5^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.17$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{12} a^{10} - \frac{1}{6} a^{9} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{24} a^{11} - \frac{1}{8} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{10} - \frac{1}{6} a^{9} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{12} a^{6} + \frac{1}{3} a^{5} - \frac{5}{12} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{24} a^{13} + \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{10} - \frac{1}{6} a^{9} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{24} a^{15} + \frac{1}{8} a^{9} - \frac{1}{6} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{72} a^{16} - \frac{1}{72} a^{15} - \frac{1}{72} a^{13} + \frac{1}{72} a^{12} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} + \frac{7}{36} a^{7} + \frac{5}{36} a^{6} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{144} a^{17} - \frac{1}{144} a^{15} + \frac{1}{72} a^{14} - \frac{1}{72} a^{12} + \frac{1}{24} a^{9} - \frac{1}{36} a^{8} - \frac{1}{4} a^{7} + \frac{1}{36} a^{6} - \frac{11}{36} a^{5} - \frac{1}{3} a^{4} + \frac{7}{18} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{288} a^{18} + \frac{1}{288} a^{16} - \frac{1}{72} a^{13} - \frac{1}{72} a^{12} - \frac{1}{24} a^{10} - \frac{2}{9} a^{9} - \frac{1}{12} a^{8} + \frac{13}{36} a^{7} + \frac{7}{24} a^{6} - \frac{1}{6} a^{5} + \frac{1}{72} a^{4} + \frac{2}{9} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{576} a^{19} + \frac{1}{576} a^{17} - \frac{1}{144} a^{16} - \frac{1}{72} a^{15} + \frac{1}{72} a^{14} - \frac{1}{48} a^{13} + \frac{1}{72} a^{12} - \frac{1}{48} a^{11} - \frac{1}{36} a^{10} - \frac{1}{8} a^{9} - \frac{1}{36} a^{8} - \frac{35}{144} a^{7} + \frac{5}{36} a^{6} - \frac{71}{144} a^{5} - \frac{5}{12} a^{4} + \frac{13}{36} a^{3} - \frac{1}{3} a^{2} + \frac{5}{12} a$, $\frac{1}{90914249373773184} a^{20} + \frac{23569802922049}{45457124686886592} a^{19} + \frac{7824484060119}{10101583263752576} a^{18} + \frac{79968008869951}{45457124686886592} a^{17} - \frac{62585022546157}{45457124686886592} a^{16} + \frac{6310813097361}{315674476992268} a^{15} + \frac{12676963982785}{2525395815938144} a^{14} - \frac{7224361619437}{1894046861953608} a^{13} - \frac{28573436785239}{2525395815938144} a^{12} - \frac{75860618760281}{11364281171721648} a^{11} + \frac{28596550674194}{710267573232603} a^{10} + \frac{227326146770309}{947023430976804} a^{9} + \frac{2651192544460753}{22728562343443296} a^{8} - \frac{2661205199199125}{11364281171721648} a^{7} + \frac{3700114237645601}{7576187447814432} a^{6} - \frac{532265898474707}{3788093723907216} a^{5} - \frac{1523460249937957}{3788093723907216} a^{4} - \frac{51708620457123}{157837238496134} a^{3} - \frac{725614469967733}{1894046861953608} a^{2} + \frac{60630976593643}{315674476992268} a - \frac{71496391778999}{315674476992268}$, $\frac{1}{13897448829272055670467297972929908172098203445288248469911658221434112} a^{21} + \frac{12157979625623107081768272048553718400524786473230359}{3474362207318013917616824493232477043024550861322062117477914555358528} a^{20} + \frac{1234905582752369704105389207582219845772108258369717595543327766915}{1544160981030228407829699774769989796899800382809805385545739802381568} a^{19} - \frac{1599873443721441630324731490866200060029429414593716295132113423657}{1737181103659006958808412246616238521512275430661031058738957277679264} a^{18} - \frac{10309468231228520647027179917693345741168233616364134842192933779923}{6948724414636027835233648986464954086049101722644124234955829110717056} a^{17} + \frac{171692195035852305983960494425418089484309838757419843268425129597}{48255030657194637744678117961562181153118761962806418298304368824424} a^{16} - \frac{22995574615913695344507983730997547364873593026339609435042656349319}{3474362207318013917616824493232477043024550861322062117477914555358528} a^{15} + \frac{28138770523724936676488421948918272720572101041111457604876195303123}{1737181103659006958808412246616238521512275430661031058738957277679264} a^{14} + \frac{2522513229871678374043559736678781034158080147576840534146318547919}{128680081752519033985808314564165816408316698567483782128811650198464} a^{13} - \frac{1740656477961756139931572359419266764696497296876292474171180420945}{434295275914751739702103061654059630378068857665257764684739319419816} a^{12} + \frac{2734686600715361663822691582873591444585636669242079075837494727353}{434295275914751739702103061654059630378068857665257764684739319419816} a^{11} - \frac{10684720758583580287959081256263019606226076693759591463233950414653}{289530183943167826468068707769373086918712571776838509789826212946544} a^{10} - \frac{778585062548580837484344522637033143295770030554187027689575136488831}{3474362207318013917616824493232477043024550861322062117477914555358528} a^{9} + \frac{20091527208353073614688409727537072027895645202010959218093797040593}{434295275914751739702103061654059630378068857665257764684739319419816} a^{8} - \frac{540234466392533123797555549587020349903329559753064958260480734529195}{1158120735772671305872274831077492347674850287107354039159304851786176} a^{7} + \frac{76859488582588617803447548935218223917404245900550664299361891323153}{434295275914751739702103061654059630378068857665257764684739319419816} a^{6} + \frac{407382760549420770244525781468896580159820059131619071412895700172567}{1737181103659006958808412246616238521512275430661031058738957277679264} a^{5} - \frac{25644202683669827655629055067990056758297304840636907804852784269943}{144765091971583913234034353884686543459356285888419254894913106473272} a^{4} + \frac{47767234567697284815392918993843047052508966511887796285315285327301}{96510061314389275489356235923124362306237523925612836596608737648848} a^{3} + \frac{891027131592191846546362281073105440157557033885228382055245134425}{24127515328597318872339058980781090576559380981403209149152184412212} a^{2} - \frac{279802288163346104804707668531881854351878017935063251199726251989}{48255030657194637744678117961562181153118761962806418298304368824424} a + \frac{654823095632871649009270657603890359120627700230111285214259885863}{2010626277383109906028254915065090881379948415116934095762682034351}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12955926374500000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T27:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15840
The 20 conjugacy class representatives for t22n27
Character table for t22n27

Intermediate fields

\(\Q(\sqrt{3}) \), 11.3.6561000000000000000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 22 sibling: data not computed
Degree 24 sibling: data not computed
Degree 44 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.8.18.63$x^{8} + 2 x^{4} + 4 x^{3} + 6$$8$$1$$18$$S_4\times C_2$$[2, 8/3, 8/3]_{3}^{2}$
2.12.24.451$x^{12} + 4 x^{11} + 4 x^{9} - 2 x^{8} + 2 x^{6} + 2 x^{4} + 4 x^{2} + 4 x - 2$$12$$1$$24$$C_2 \times S_4$$[2, 8/3, 8/3]_{3}^{2}$
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.10.18.2$x^{10} + 10 x^{8} + 40 x^{6} + 60 x^{5} + 80 x^{4} - 75 x^{3} + 80 x^{2} + 75 x + 57$$5$$2$$18$$F_{5}\times C_2$$[9/4]_{4}^{2}$
5.10.18.2$x^{10} + 10 x^{8} + 40 x^{6} + 60 x^{5} + 80 x^{4} - 75 x^{3} + 80 x^{2} + 75 x + 57$$5$$2$$18$$F_{5}\times C_2$$[9/4]_{4}^{2}$